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On velocity-dependent potentials in quantum mechanics

Authors
Journal
Nuclear Physics B
0550-3213
Publisher
Elsevier
Publication Date
Volume
16
Issue
2
Identifiers
DOI: 10.1016/0550-3213(70)90258-0
Disciplines
  • Physics

Abstract

Abstract Velocity-dependent potentials are investigated in both the Lagrangian and the Hamiltonian formalism in quantum mechanics. In order to achieve a consistent method, a canonical transformation is introduced for the type of Lagrangian 1 2 q ig ij(q) q j − V(q) where g ij and V are functions of position operators q i only. It is found that the proper Hamiltonian satisfying the canonical equation of motion should be H = p iq i − L − Z where Z is a function of q i and is expressed in terms of g ij ( q). This formulation has been examined by some examples. The Euler-Lagrange equation that is consistent with the canonical equation of motion is derived and it turns out to be in an apparently dissipated form. However, the physical system could be a non-dissipative one. Finally, the Schroedinger equation is investigated according to the above argument.

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